∆Ms theory precision confronts flavour anomalies

Based on recent HQET sum rule and lattice calculations we present updated Standard Model predictions for the mass differences of neutral B mesons: ΔMsSM=18.4−1.2+0.7ps−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varDelta {M}_s^{\mathrm{SM}}=\left({18.4}_{-1.2}^{+0.7}\right){\mathrm{ps}}^{-1} $$\end{document} and ΔMdSM=0.533−0.036+0.022ps−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varDelta {M}_d^{\mathrm{SM}}=\left({0.533}_{-0.036}^{+0.022}\right){\mathrm{ps}}^{-1} $$\end{document} study their impact onnew physics models that address the present hints of anomalous data in b → sℓℓ transitions. We also examine future prospects of further reducing the theory uncertainties and discuss the implications of a 2025 scenario with ΔMsSM2025=18.4±0.5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varDelta {M}_s^{\mathrm{SM}\;2025}=\left(18.4\pm 0.5\right) $$\end{document} ps−1. In particular, the latter yields upper bounds MZ′ ≲ 9TeV and MS3≲30TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {M}_{S_3}\lesssim 30\mathrm{TeV} $$\end{document}for the minimal Z′ and S3 lepto-quark explanations of the b → sℓℓ anomalies, respectively.


Introduction
The mixing of neutral B mesons provides an important test of our understanding of the Standard Model (SM). It gives direct access to the poorly known CKM elements V td , V ts and V tb , and it is an excellent probe for physics beyond the standard model (BSM), see e.g. [1].
By now the experimental values of the mass differences of the neutral B mesons are known very precisely [2] (based on the measurements in [3][4][5][6][7][8][9][10]) The theoretical determination of the mass differences is limited by our understanding of non-perturbative matrix elements of dimension six operators. The matrix elements can be determined with lattice simulations or sum rules. Both approaches utilize a three-point correlator of two interpolating currents for the B q andB q mesons and the dimension six operator, which is significantly more complicated than the determination of decay constants where one considers a two-point correlator of two currents. Therefore, for a long time (see e.g. the FLAG 2013 review [11]) the precision of the matrix elements was limited to more JHEP12(2019)009 than 10%. The consequence were SM predictions of ∆M q [1] close to the experimental numbers, but with much higher uncertainties: In 2016 [12] the FNAL/MILC collaboration presented a new evaluation of the nonperturbative B mixing matrix elements yielding larger values than previously obtained. They achieved relative errors of 8.5% and 6.4% for the matrix elements relevant for B d and B s mixing in the SM, respectively, thus pushing the uncertainty of simulations with 2+1 dynamical flavours below the 10% benchmark for the first time. The FNAL/MILC results dominate the current FLAG average [13] and we get the following predictions 1 which, when combined, are about two standard deviations above the experimental numbers. The values in eq. (1.5) and eq. (1.6) for the mass differences could be the starting point of new anomalies arising in the b-sector, see e.g. [14][15][16][17][18]. In addition ∆M s poses very severe constraints on certain BSM models -in particular Z models -that address the observed anomalies in b → s transitions, see e.g. [19].
To date several new determinations of the matrix elements with a precision below 10% have appeared: first from sum rules [20][21][22][23][24] and very recently from the HPQCD collaboration [25]. The SU(3) breaking ratios were also evaluated by the RBC/UKQCD collaboration [26]. We compare the various predictions in figure 1 where also the results using a weighted average of the values of [13,[23][24][25][26] for the matrix elements are shown. The weighted average gives an impressive leap of the precision to 3.6% and 3.1% for the matrix elements for B d and B s mixing. We observe excellent agreement between the sum rule prediction and the weighted average and given that lattice simulations and sum rules have very different systematic uncertainties this is a strong confirmation of both methods. We discuss the status of the non-perturbative input in more detail in section 2. With respect to eq. (1.5) and eq. (1. 6

∆M s in the Standard Model
In the SM, B s mixing is generated by the diagrams shown in figure 2. The observable of interest in this work is the mass difference of the two mass eigenstates: The SM calculation (see e.g. [1] for a review) gives the following result for M s 12 M s, SM where λ t = V tb V * ts , S 0 is the Inami-Lim function [29], x t = (m t (m t )/M W ) 2 andη B ≈ 0.84 encodes the 2-loop perturbative QCD corrections [30]. In the SM, only a single four-quark ∆B = 2 operator arises, 3) whose hadronic matrix element is parameterised in terms of the meson decay constant f B s and a bag parameter B 1 Note that we have used a form of eq. (2.2) (matching that in [1]) where the bag parameter B 1 depends on the b quark scale µ b ∼ m b . An alternative notation (commonly used by lattice groups, for example see FLAG [13]) uses the pair (η B ,B 1 ) instead of (η B , B 1 (µ b )), withB 1 a renormalization group (RG) invariant quantity. The two are related such that their product is equal, and explicitly we havê The combination f 2 B s B 1 is the least-well known parameter for B s mixing, and the most important as the observable ∆M s is directly proportional to it. In the SM determination of the decay rate difference ∆Γ q and for BSM contributions to ∆M q in addition to Q 1 four more operators arise: which are typically parameterised as  . We show the ETMC '13 values in orange, the FLAG 2+1 average [13] for the operator Q 1 together with the FNAL/MILC '16 results [12] for the other operators in red, the sum rule values [24] in blue and the HPQCD '19 results [25] in cyan. The gray rectangles indicate the weighted averages of the first three determinations.
The full basis of matrix elements has been determined with lattice simulations [12,25,31] and sum rules [23,24]. In [12] the combinations f 2 B s B i were determined and combined with the 2016 PDG average for the decay constants to extract the Bag parameters. This implies that the latter suffer from bigger uncertainties because one cannot exploit partial cancellations of uncertainties which occur when f B s is determined in the same simulation. All other works determine the Bag parameters B i (µ b ) as their central results. This has the advantage that the values can easily be combined with updated results for the decay constant f B s which has seen significant improvements in the last few years and we use the FLAG 2+1+1 average [13] of the results [32][33][34] below.
In fact the sum rule method directly determines the Bag parameters and requires an independent result for the decay constant to obtain the matrix elements. In this approach one has to determine the perturbative 3-loop contribution to a correlator describing B mixing. Based on the master integrals presented in [35] this calculation was first performed for B d mixing by [20][21][22] and confirmed in [23], where this method was also extended to different mixing operators and to lifetime operators. In [24] corrections due to a finite value of the strange quark mass were determined, yielding new predictions for the mass differences in better agreement with experiment than the FNAL/MILC values (eqs. (1.5) and (1.6)) with similar precision.
Very recently an independent lattice study was performed by HPQCD [25] which does not confirm the large FNAL/MILC predictions for the mass differences. We present a comparison of the combinations f 2 B s B i in figure 3. Besides the individual results [12,13,24,25,31] we show the weighted averages thereof where we have excluded [31]  The case of f 2 B s B 3 is similar to that of the SM matrix element but the uncertainties are considerably larger. Finally for f 2 B s B 4 and f 2 B s B 5 we find a discrepancy between the results from ETMC '13 and the remaining ones. This difference might result from the use of RI-MOM renormalisation scheme in [31], see e.g. the discussion in [36] for a similar problem in the K sector. The other values are in good agreement with each other, with KLR '19 yielding a somewhat larger result than the lattice simulations for f 2 B s B 5 . Our weighted averages at the scale For convenience we also provide the weighted averages for B d mixing and the Bag parameters in A. The ratio of the mass differences of B s and B d mesons benefits from the cancellation of many uncertainties and in the SM we get of the values from [13,[24][25][26] is dominated by the sum rule result [24] which is about a factor of two more precise than the lattice determinations. Using the CKMFitter result  This allows a very precise determination of |V td /V ts | from B mixing [24]. We note that the product of (2.13) and (1.8) is about 3% larger than the value for ∆M d from a direct determination (1.7). The reason is that we also include the RBC/UKQCD '18 [26] result in the average (2.10) for ξ but not in the averages for the individual matrix elements since those were not determined there. 2 The ellipses in figure 1 use f 2 B s B 1 and ξ as inputs and the ellipse for the weighted average therefore also gives a value of ∆M d which is about 3% larger than (1.7). We further note that our ellipses for the FLAG '19 and HPQCD '19 results differ from the numerical results given in [12] and [25] because we use different values for the CKM parameters as discussed in A.
All in all we observe a consistent picture of different non-perturbative determinations for B mixing. The future (∼ 2025) prospects for lattice simulations of the matrix elements have been assessed in [27,37]. The relative uncertainties for f 2 B s B 1 are estimated as 3% [27] and 1.3% [37]. Given that the two latest lattice results [12,25] do not overlap within their uncertainties and differ by 15% we believe the latter estimate to be overly optimistic and assume the scenario of [27]. On the other hand the uncertainties in sum rule analyses are dominated by the one from the matching between QCD and Heavy Quark Effective Theory (HQET) matrix elements which can be reduced significantly by a perturbative twoloop matching calculation [21,22] or eliminated altogether by using QCD [38,39] instead of HQET sum rules. We expect that a reduction of the sum rule uncertainty for B 1 to 3% is realistic. The effect on the weighted average is a reduction of the error from 3.1% to about 2%.
Simultaneously, new experimental data from Belle II and the LHC will improve our knowledge of the CKM element V cb which constrains V ts in the CKM unitarity fits and is the other critical input to our theory predictions. The Belle II collaboration forecasts [27] that by 2025 the uncertainty on V cb will be at the level of 1%. Using these future predictions

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Parameter Future value (our explicit future inputs are given in table 1, with all others staying the same as at present), we expect that ∆M s will be predicted with an uncertainty of only ±0.5 ps −1 by 2025, yielding eq. (1.9). 3 3 ∆M s interplay with flavour anomalies

Status and future prospects of the b → s anomalies
A key application of the refined SM prediction for ∆M s is in the context of the recent hints of lepton flavour universality (LFU) violation in semi-leptonic B meson decays. Focussing on neutral current anomalies, the main observables are the LFU violating ratios [40][41][42], together with the angular distri- [43][44][45][46][47][48][49][50][51][52] and the branching ratios of hadronic b → sµ + µ − decays [43,44,53]. The effective Lagrangian for semi-leptonic b → sµ + µ − transitions contains the terms and the primed operators O µ 9,10 having the opposite (L → R) quark chiralities. Intriguingly, one obtains a very good description of the b → s data by allowing new physics only in a single combination of Wilson coefficients (for recent fits see e.g. [54][55][56][57][58][59][60]). The bestfit scenario δC µ 9 = −δC µ 10 ≈ −0.53 ± 0.08 of [54] corresponds to an effective operator with left-handed (LH) quark and muon currents and we consider its interplay with ∆M s in section 3.4. The data is also well represented by assuming either δC µ 9 or δC µ 10 being different from zero, corresponding to vector or axial-vector muon currents, respectively. However, we find that the mixing constraints on these scenarios are stronger than for case of LH muon currents, since larger values of the Wilson coefficients are required, and do not discuss them further. Following the update of R K from LHCb [42], there has been an increased favourability of right handed (RH) current contributions in the quark sector and we investigate their effect in section 3.5.2.

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By 2025 LHCb expects a serious improvement in the measured precision on R K and R K * compared to the situation today -they forecast [28] the error on R K will be reduced to ±0.025 (better than a factor of two compared to the error reported at Moriond 2019) and to ±0.031 for R K * (a factor of four better). If these precisions are indeed realised, and the central values stay close to their currently measured values, LHCb will have made a discovery of LFU violation at the level of 6 σ and 10 σ for R K and R K ( * ) respectively.

New physics contributions to ∆M s in the effective theory
The NP contributions to B s mixing can be described by the effective Lagrangian In general, one would expect that any NP contributing to b → s transitions will eventually feed into B s mixing. However, this connection is hidden at the effective operator level where the Wilson coefficients in eq. (3.1) and eq. (3.5) are independent. 5 It is hence crucial to focus on specific UV realizations in order to explore the connection between b → s anomalies and ∆M s and we will consider two simplified models below. 4 This expression neglects the top-quark threshold, which is a sub-percent effect. However, the correct running is taken into account in our numerics and figures. There are two basic possibilities how the effective operators in eq. (3.1) can be generated at tree level: by the exchange of a Z mediator coupling to the quark and the muon current or by the exchange of a lepto-quark coupling to mixed quark-muon currents. Anticipating our results from section 3.4 we describe the Z scenario in detail below. In addition we consider the case of the scalar lepto-quark S 3 ∼ (3, 3, 1/3), whose defining Lagrangian can be found e.g. in [19,62].
We consider a simplified Z model with only those couplings required to explain the observed flavour anomalies, with the Lagrangian where d i and i label the different generations of down-type quark and charged lepton mass eigenstates respectively, and λ Q,d,L are hermitian flavour space matrices. We have neglected RH currents in the lepton sector in the above eq. (3.7) since the latter actually worsen the compatibility with ∆M s , as they require a larger Wilson coefficient.
Integrating out the Z at tree level, we get the effective Lagrangian The first line contains the terms that contribute to the rare decay coefficients C ( )µ 9,10 , the second a contribution to ∆M s through the same operator as in the SM, and the third contributions to ∆M s from operators that do not appear in the SM.
Matching onto the effective Lagrangians for the low energy observables in eqs. (3.1) and (3.4), we find (at the scale µ = M Z ) (3.13)

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Inserting the above expressions in eq. (3.5), using our combined averages for the bag parameters, and taking a typical Z scale of 5 TeV we find that the contribution to ∆M s can be written as 6 which shows a significant RG enhancement for the LR operator. Besides B s mixing, we consider a constraint that is particularly relevant for light Z interacting with muons via LH currents. By SU(2) L invariance the Z couples also to LH neutrinos via the λ L 22 coupling which is required by the b → sµµ anomalies. Hence, one has an extra term in the Z effective Lagrangian of the type which leads to the neutrino trident production, ν µ N → ν µ µ + µ − N . Using the most recent calculation for the SM cross-section, one gets [63] σ SM+NP (with v ≈ 246 GeV and s 2 W ≈ 0.231), which is constrained by the existing CCFR measurement σ CCFR /σ SM CCFR = 0.82 ± 0.28 [64]. This result implies M Z /λ L 22 > 0.47 TeV at 2 σ. The upcoming DUNE experiment [65] is expected to also measure this process, however the precision it will achieve (see e.g. [63]), suggest its data will not increase much the limits on this parameter combination and so we will use the CCFR bound throughout.
Last but not least, the parameter space is constrained by direct LHC searches [66,67] and perturbative unitarity [68,69]. Whether a few TeV Z is ruled out by LHC direct searches crucially depends on the details of the Z model. The stringent constraints from di-lepton searches [66] are tamed in models where the Z does not couple to valence quarks. For instance, assuming only the two couplings required by the b → s anomaly, namely λ Q 23 and λ L 22 , one finds that current Z searches are not sensitive yet (see e.g. [70,71]). Assuming the best-fit scenario δC µ 9 = −δC µ 10 ≈ −0.53 ± 0.08 [54] is generated by the exchange of a Z we obtain the perturbative unitarity bound M Z < 59 TeV [68]. The same limit for the lepto-quark scenario reads M S 3 < 69 TeV [68]. The mass of the lepto-quark is also bounded from below up to about 1 TeV [67] from direct searches at the LHC. and real couplings (these two assumptions will be relaxed in section 3.5). We obtain the predictions

∆M s interplay with flavour anomalies in the minimal scenarios
for the lepto-quark.
In figure 5 we compare the new physics contributions ∆M NP s in the two models to the allowed two-sigma regions for the present and future scenarios. The dashed lines show what effect is predicted for a given value of the Wilson coefficient combination δC µ 9 = −δC µ 10 , whereby the envelope of these lines corresponds to the two-sigma region from the fit JHEP12(2019)009 δC µ 9 = −δC µ 10 = −0.53 ± 0.08 to the present b → s data. In addition we show constraints from neutrino trident production, direct searches by the LHC experiments and perturbative unitarity as hatched regions.
The left plot shows the simplified Z model where the relation between ∆M s and δC 9 depends on the parameter combination M Z /λ L 22 . We observe that a Z explanation of the b → s data causes a sizeable positive contribution to ∆M s unless the Z mass is rather light. Values of ∆M NP s outside the shaded regions are in tension with the respective predictions for ∆M s at the level of two standard deviations. This allows us to place upper bounds on the mass of the Z . 7 For instance, we find that the large central value of the FLAG '19 prediction in eq. (1.6) (green region) requires M Z 4 TeV for λ L 22 = 1 in order to explain b → s at 1 σ. The limit is about a factor 3 /2 weaker in the case of the Average '19 value of ∆M s (purple region) despite the reduction of the uncertainty by over 30% with respect to FLAG '19 because the central value is closer to the experimental result. It is straightforward to rescale the bound for different lepton couplings. The combination of ∆M s and perturbative unitarity (requiring λ The lepto-quark model is shown in the right plot and yields smaller new physics contributions to ∆M s because the effect is only generated at loop level. Using the Average '19 value (purple) we find that a leptoquark explanation is still viable in the entire mass range between implies an upper limit of about 30 TeV on the lepto-quark mass M S 3 , better than the perturbativity bound by more than a factor of two.
We conclude that both simplified Z and lepto-quark models currently provide possible solutions to the b → s anomalies, with the previous stringent constraints [19] from the FLAG '19 values for ∆M s being relaxed by the refined SM prediction Average '19. However, with the future increase in the precision of ∆M s lepto-quark explanations will be favoured over a Z , assuming that the central values remain similar. In section 3.5 we therefore discuss ideas how the implied tension might be reduced by generalizations of the minimal Z scenario.

New physics options beyond the minimal Z scenario
The constraints on the minimal Z scenario considered above are particularly strong because the latter predicts a positive contribution to ∆M s , while the central values of the current SM predictions are already larger than the experimental result. Therefore, we will explore two simple options for how a negative contribution to ∆M s could be generated, thus relaxing the tension. Examining eq. (3.14) the two cases present themselves: 1. For only LH quark coupling (i.e. λ d = 0), a negative contribution can only be generated if λ Q 23 acquires a complex phase.
2. With both LH and RH quark couplings, real couplings alone can give the required sign if the interference term is large enough.

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Although these two possibilities were already identified in [19], their compatibility with global fits for flavour anomalies was only partially assessed in [72]. We start with CP violating couplings in section 3.5.1, and then study the case for RH quark currents in section 3.5.2. As a benchmark point, we take a lepton coupling of λ L 22 = 1 and a Z mass of 5 TeV where the best-fit result δC µ 9 = −δC µ 10 = −0.53 saturates the two-sigma range of the Average '19 prediction (see figure 5). In the following, we address the question whether either option 1. or 2. allows for this benchmark point to be viable within the FLAG '19 or Future '25 scenarios.

CP violating couplings
It is clear from eq. (3.14) that if λ Q 23 has a large enough imaginary part, then it will be possible to bring down the theory prediction for ∆M s below the SM prediction and into better agreement with experiment. However, once we allow for complex Z quark couplings, there are extra constraints to be considered, in the form of CP-violating observables that arise from B s mixing. The most relevant here is the mixing-induced CP asymmetry [1,73], arising from interference between B meson mixing and decay. The semi-leptonic CP asymmetries for flavour-specific decays, a s sl , are not competitive yet since the experimental errors are still too large [1]. Defining (with η as in eq. (3.5)) the mixing-induced CP asymmetry is given by where A mix CP = −0.021 ± 0.031 [2,74], β s = 0.01843 +0.00048 −0.00034 [75], and we have neglected penguin contributions [1].
New phases in λ Q 23 also imply a complex value for δC µ 9 , which has not typically been considered in previous global fits (see [76,77] for exceptions). We perform our own fit using the flavio software [78], using the same set of b → s observables as [79].
The result of our fit is shown in figure 6, with the higher SM prediction ∆M FLAG '19 s corresponding to the green bands. There is no overlap between the 1 σ regions for our three observables, since A mix CP provides too strong a constraint on the imaginary part of the coupling. So a complex coupling does not provide a way to significantly evade the strong bounds from ∆M s . Using instead our prediction ∆M Future '25 s (orange) there is still no overlap, for the same reason. As such, the addition of complex phases to the Z quark coupling does not alleviate tension at the benchmark point in the Future '25 scenario, which is important given the strength of this future bound, as shown in figure 5.

Right-handed couplings
A Z coupling to a RH quark current can also allow for a negative BSM contribution to ∆M s through the interference term, which has the advantage that it is RG enhanced by roughly an order of magnitude (see eq. (3.14)) relative to the pure LL or RR operators. In our JHEP12(2019)009 previous work [19,72] we stated that RH quark currents break the approximate symmetry R K ≈ R K * and hence were disfavoured by the data. On the other hand, the recent update of R K at Moriond 2019 [42] favours a non-zero RH quark current contribution, as widely discussed in the literature since then (see e.g. [54][55][56][57][58][59][60]). There is however a problem with this solution, which is that in our Z model, at the leading order in the NP contribution, the behaviour can be written as where p ≈ 0.86 is the polarization fraction [80]. The current experimental measurements suggest (R K − 1)/(R K * − 1) ≈ 0.50, which requires λ d 23 /λ Q 23 ≈ −0.37. On the other hand, eq. (3.14) shows that the same sign for LH and RH quark couplings are needed to reduce ∆M s . This is evident in our fit result shown in figure 7, where the allowed region for the b → s anomalies is partially driven by the R K ( * ) measurements. With the high prediction ∆M FLAG '19 s , there is a clear gap between the 1 σ allowed regions, which does not allow us to solve the tension for the benchmark point (for a similar conclusion see also [58]). If we instead examine the orange region, corresponding to the ∆M Future '25 s prediction, 8 the gap has shrunk by only a small amount, and so our benchmark point is only marginally more favoured.

Beyond simplified Z models
Within the simplified Z model considered so far, neither of the two options above (imaginary couplings and RH quark currents) helps to improve the compatibility between the b → s measurements and the expected precision determination value of ∆M s in 2025. There are, however, other possibilities in order to achieve that when going beyond our simplified Z setup. For instance, sticking only to R K and R K * (so no angular observables, etc.), these can be accommodated via NP in electrons featuring sizeable contributions both from LH and RH quark currents, so that a negative contribution to ∆M s is possible as well. This class of models were suggested in the context of Composite Higgs scenarios [81]. Alternatively, as pointed out in [82], in UV complete models of the vector lepto-quark U µ ∼ (3, 1, 2/3) [83][84][85][86][87][88] addressing both R D ( * ) and b → s anomalies, the couplings to quarks of extra Z and/or coloron states (not directly responsible for the anomalies in semi-leptonic B decays) can naturally have a large phase in order to accommodate a negative ∆M s , without being in tension with CP violating observables. These two examples show that although it is difficult to obtain ∆M NP s < 0 within simplified models for the b → s anomalies, that is certainly not impossible in more general constructions.

Conclusions
In this paper, we have combined recent results from sum rules [23,24] (by some of the current authors, see also [20][21][22] for consistent results for the operator Q 1 by a different group), and the FNAL/MILC [12] and HPQCD [25]  These results are about 40% more precise than the values in eq. (1.5) and eq. (1.6) obtained from the current FLAG [13] averages and show better agreement with the experimental results. The average for the SU(3)-breaking ratio ξ that determines the SM predictions for ∆M d /∆M s is dominated by the SR calculation combined with the most recent lattice results for the decay constants, which provides a precision much better than the latticealone value for this quantity. The new averages agree very well with the individual SR results (see figure 3 and figure 4), further reinforcing the usefulness of this as a totally independent determination of these important non-perturbative parameters. With our new average the hadronic uncertainties in these quantities are now at the same level as the CKM uncertainties. We then argued that with improved determinations of the hadronic matrix elements and the CKM elements the precision in ∆M s can be improved to 3% by about 2025. We investigated the constraints from ∆M s on minimal Z and LQ explanations of the b → s anomalies. The constraints for LQs are generally weaker because contributions to B s mixing are only generated at loop level. Still we find that the assumed Future '25 scenario leads to an upper bound of 30 TeV on the LQ mass, stronger than the unitarity constraint [68,69] by more than a factor of two. The situation for the Z is summarized in figure 8. Despite the reduction of the uncertainty the constraints with the updated prediction Average '19 are weaker by about a factor 3 /2 than those obtained with the FLAG '19 scenario like in our earlier work [19] because of the smaller central value. Nevertheless, B s mixing by itself is sufficient to exclude the minimal Z scenario with a benchmark value of λ L 22 = 1 for the lepton coupling for Z masses above 6 TeV. Assuming the Future '25 scenario, we obtain the upper limit M Z 9 TeV for lepton couplings saturating the perturbativity bound (see right panel in figure 8), which is about six times better than the unitarity bound on the Z mass. On the other hand, the left plot in figure 8 demonstrates that the parameter space will be very strongly constrained for more perturbative values of λ L 22 . An interesting open question is to what extent this mass window can be covered by direct searches at different future colliders.
Last but not least, we have addressed the question whether an extension of the minimal Z scenario might relax the strong constraints. We have discussed two possibilities in which the minimal model might be extended, by adding new CP violating phases to the quark coupling, or including a coupling to both LH and RH quarks. For the case of new CP phases, the B s mixing observable A CP mix is very strongly constraining unless the real part is sufficiently small, which prevents an explanation of the anomalies. Adding instead a coupling to RH quarks has been recently discussed in the literature following the update, earlier this year at Moriond, of the LHCb measurement of R K . We demonstrate that the currently observed pattern, with R K * < R K < 1 requires a particular sign combination for the LH and RH quark couplings, which would lead to a positive shift to ∆M s rather than a JHEP12(2019)009 reduction. We conclude that the constraints from B s mixing cannot be easily avoided within this class of minimally extended Z simplified models, although more general constructions can achieve that.
All in all, we have shown how we are now entering the age of precision determinations of ∆M s and that, as we proceed further, the latter will become an increasingly powerful tool in order to constrain new physics explanations of other flavour observables.  We observe that this result is very close to that from the standard fit shown in table 2 and therefore use the standard fit for simplicity. In addition to our averages given in eq. (2.8) we also provide the weighted averages for the matrix elements in the B d system